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 \begin{center} \textbf{Math 475 Homework 7} \\
 \textsc{Due April 15th, 2009} \end{center}
 \begin{enumerate}
 \item Find a quadratic function that models this data: \\[.1in] \begin{tabular}{c | c} IN & OUT \\ \hline 0 & -6 \\ 1 & -1 \\ 2 & -6 \\ 3 & -21 \\ 4 & -46 \\ 5 & -81 \\ \end{tabular} \item Prove that the first difference of an $N$th degree polynomial has at most degree $N-1$.
 \item Prove an $N$th degree polynomial has constant $N$th differences. (Hint: use the last answer.)
 \item The $n$th tetrahedral number is the sum of the first $n$ triangular numbers. (It's also the balls in first $n$ layers of a tetrahedral ball pyramid.) Find a closed formula for it using our analysis of functions and their first, second, third, etc. differences.
 \item Write me a formula for sums of cubes using our difference table analysis.
 \item Derive our class solution for $x^2 + Bx + C = 0$, then explain how you can solve ANY quadratic equation using this special case and derive the Quadratic Formula.
 \end{enumerate}
 
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